Length Of Horizon Research Articles

A distributionally robust stochastic optimization-based model predictive control with distributionally robust chance constraints for cooperative adaptive cruise control under uncertain traffic conditions

Motivated by connected and automated vehicle (CAV) technologies, this paper proposes a data-driven optimization-based Model Predictive Control (MPC) modeling framework for the Cooperative Adaptive Cruise Control (CACC) of a string of CAVs under uncertain traffic conditions. The proposed data-driven optimization-based MPC modeling framework aims to improve the stability, robustness, and safety of longitudinal cooperative automated driving involving a string of CAVs under uncertain traffic conditions using Vehicle-to-Vehicle (V2V) data. Based on an online learning-based driving dynamics prediction model, we predict the uncertain driving states of the vehicles preceding the controlled CAVs. With the predicted driving states of the preceding vehicles, we solve a constrained Finite-Horizon Optimal Control problem to predict the uncertain driving states of the controlled CAVs. To obtain the optimal acceleration or deceleration commands for the CAVs under uncertainties, we formulate a Distributionally Robust Stochastic Optimization (DRSO) model (i.e. a special case of data-driven optimization models under moment bounds) with a Distributionally Robust Chance Constraint (DRCC). The predicted uncertain driving states of the immediately preceding vehicles and the controlled CAVs will be utilized in the safety constraint and the reference driving states of the DRSO-DRCC model. To solve the minimax program of the DRSO-DRCC model, we reformulate the relaxed dual problem as a Semidefinite Program (SDP) of the original DRSO-DRCC model based on the strong duality theory and the Semidefinite Relaxation technique. In addition, we propose two methods for solving the relaxed SDP problem. We use Next Generation Simulation (NGSIM) data to demonstrate the proposed model in numerical experiments. The experimental results and analyses demonstrate that the proposed model can obtain string-stable, robust, and safe longitudinal cooperative automated driving control of CAVs by proper settings, including the driving-dynamics prediction model, prediction horizon lengths, and time headways. Computational analyses are conducted to validate the efficiency of the proposed methods for solving the DRSO-DRCC model for real-time automated driving applications within proper settings.

Dynamic lot-sizing model under perishability, substitution, and limited storage capacity

Various commodities including blood, pharmaceutical, and agricultural products are perishable in nature and require special storage conditions, such as controlled temperature. However, expanding the space of cold storage is expensive. Therefore, storage capacity frequently restricts the operational efficiency of enterprises in the perishable product industry. Numerous enterprises have adopted the policies of product substitution and multiple item joint procurement to achieve an efficient inventory management and reduce their operation costs. This paper considers a two-product dynamic lot-sizing problem for perishable inventory under product substitution and limited storage capacity. This study aims to identify the ordering, inventory, and substitution decisions over a planning horizon under the criteria that (i) the inventory holding costs and deterioration rates are age dependent, (ii) a one-direction product substitution and joint ordering of two products are possible, and (iii) the storage capacity has an upper bound that limits the inventory quantities. The contributions of this study are summarized in three points. First, we develop a dynamic programming algorithm by using two structural properties to solve the dynamic lot-sizing problem with perishable inventory and demand substitution under storage capacity constraints. Second, we obtain the forecast horizons for the general problem by using the marginal analysis method and for the case with constant unit ordering costs through establishing the monotonicity of the regeneration points of two products. Third, we determine the effects of storage capacity, product lifetime, joint setup, and inventory costs on the length of the forecast horizon and the total costs by using a detailed test bed of instances. The major findings are also discussed in three aspects. First, the single-period satisfaction property fails to hold under limited storage capacity and time-varying unit ordering costs. Second, the forecast horizon inconsistently increases with joint setup costs or storage capacity. Third, the total costs fail to decrease with increasing storage capacity or product lifetime.

Sensitivity Analysis to Define Guidelines for Predictive Control Design

Signalized traffic control is important in traffic management to reduce congestion in urban areas. With recent technological developments, more data have become available to the controllers and advanced state estimation and prediction methods have been developed that use these data. To fully benefit from these techniques in the design of signalized traffic controllers, it is important to look at the quality of the estimated and predicted input quantities in relation to the performance of the controllers. Therefore, in this paper, a general framework for sensitivity analysis is proposed, to analyze the effect of erroneous input quantities on the performance of different types of signalized traffic control. The framework is illustrated for predictive control with different adaptivity levels. Experimental relations between the performance of the control system and the prediction horizon are obtained for perfect and erroneous predictions. The results show that prediction improves the performance of a signalized traffic controller, even in most of the cases with erroneous input data. Moreover, controllers with high adaptivity seem to outperform controllers with low adaptivity, under both perfect and erroneous predictions. The outcome of the sensitivity analysis contributes to understanding the relations between information quality and performance of signalized traffic control. In the design phase of a controller, this insight can be used to make choices on the length of the prediction horizon, the level of adaptivity of the controller, the representativeness of the objective of the control system, and the input quantities that need to be estimated and predicted the most accurately.

Optimal Power Allocation in Battery/Supercapacitor Electric Vehicles Using Convex Optimization

This paper presents a framework for optimizing the power allocation between a battery and supercapacitor in an electric vehicle energy storage system. A convex optimal control formulation is proposed that minimizes total energy consumption whilst enforcing hard constraints on power output and total energy stored in the battery and supercapacitor. An alternating direction method of multipliers (ADMM) algorithm is proposed, for which computational and memory requirements scale linearly with the length of the prediction horizon (and can be reduced using parallel processing). The optimal controller is compared with a low-pass filter against an all-battery baseline in numerical simulations, where it is shown to provide significant improvement in battery degradation (inferred through reductions of 71.4% in peak battery power, 21.0% in root-mean-squared battery power, and 13.7% in battery throughput), and a reduction of 5.7% in energy consumption. It is also shown that the ADMM algorithm can solve the optimization problem in a fraction of a second for prediction horizons of more than 15 minutes, and is therefore a promising candidate for online receding-horizon control.

Self-Tuning Unbiased Finite Impulse Response Filtering Algorithm for Processes With Unknown Measurement Noise Covariance

An unbiased finite impulse response (UFIR) filtering algorithm is designed in the discrete-time state-space for industrial processes with unknown measurement data covariance. By assuming an inverse-Wishart distribution, the data noise covariance is recursively estimated using the variational Bayesian (VB) approach. The optimal averaging horizon length N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">opt</sub> is estimated in real time by incorporating the estimated data noise covariance into the full-horizon UFIR filter and specifying N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">opt</sub> at a point, where the estimation error covariance reaches a minimum. The proposed VB-UFIR algorithm is applied to a quadrupled water tank system and moving target tracking. It is demonstrated that the VB-UFIR filter self-estimates N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">opt</sub> more accurately than known solutions. Furthermore, the VB-UFIR filter is not prone to divergence and produces more stable and more reliable estimates than the VB-Kalman filter.

Coordinated management of connected plug-in hybrid electric buses for energy saving, inter-vehicle safety, and battery health

For plug-in hybrid electric vehicles, deeper battery discharge provides more electrical energy at a lower cost than fossil fuel, which reduces the overall energy consumption cost. However, it also accelerates battery degradation and increases the equivalent cost of battery life loss. Therefore, the battery depth of discharge (DOD) needs to be optimized to minimize the overall cost. For connected plug-in hybrid electric vehicles, the longitudinal velocity planning determines the energy demands, which directly affects the charging or discharging current to the battery and therefore affects DOD, aging, and fuel consumption as well. For connected plug-in hybrid electric buses running on fixed routes, in order to coordinate the velocity planning and battery health protection, this paper proposes a real-time energy management strategy aimed at achieving the minimum overall cost by optimizing the DOD and velocity planning. The proposed method is evaluated in an urban traffic scenario, and the goal of achieving optimal DOD is divided into a co-optimization problem over each moving horizon, where the velocity planning and energy management are traded off by minimizing the sum of driving safety cost, energy consumption cost, and equivalent cost of battery life loss. The results show that the proposed far-sighted economy-oriented methodology is superior to a short-sighted velocity planning and energy management method, and has an obvious advantage in the total cost compared with other conventional methods using a preset DOD. Moreover, the impacts of possible communication delays and prediction horizon lengths on the optimization performance and computational cost are investigated. The proposed method provides a promising management strategy for future connected and autonomous mobility design, which can mitigate battery capacity degradation and improve the fuel economy.

Finite-Horizon Discounted Optimal Control: Stability and Performance

Motivated by (approximate) dynamic programming and model predictive control problems, we analyse the stability of deterministic nonlinear discrete-time systems whose inputs minimize a discounted finite-horizon cost. We assume that the system satisfies stabilizability and detectability properties with respect to the stage cost. Then, a Lyapunov function for the closed-loop system is constructed and a uniform semiglobal stability property is ensured, where the adjustable parameters are both the discount factor and the horizon length, which corresponds to the number of iterations for dynamic programming algorithms like value iteration. Stronger stability properties such as global exponential stability are also provided by strengthening the initial assumptions. We give bounds on the discount factor and the horizon length under which stability holds and we provide conditions under which these are less conservative than the bounds of the literature for discounted infinite-horizon cost and undiscounted finite-horizon costs, respectively. In addition, we provide new relationships between the optimal value functions of the discounted, undiscounted, infinite-horizon and finite-horizon costs respectively, which are very different from those available in the approximate dynamic programming literature. These relationships rely on assumptions that are more likely to be satisfied in a control context. Finally, we investigate stability when only a near-optimal sequence of inputs for the discounted finite-horizon cost is available, covering approximate value iteration as a particular case.

Learning the Best Price and Ordering Policy under Fixed Costs and Ambiguous Demand

We study joint inventory-price control in which a firm chooses among a finite number of prices to influence the demand to be realized; also, the firm’s ordering activities incur fixed setup costs. While intending to settle down on an optimal price and figure out an optimal ordering policy all catering to the long-run average criterion, the firm is ambiguous about the stationary distribution of the random demand that it is to face under each price. We propose an adaptive policy in which periods are grouped into intervals, with each being associated with one single price and one single ordering policy. Pricing is based on a learning-while-doing trade-off: a price with the least number of interval visits will be chosen when this number is below a threshold associated with the total number of interval visits under all prices; otherwise, the chosen price will be one with the most promising profit prospect estimated from past experiences. Interval-wise ordering relies on an (s,S) policy most suitable for the empirical distribution learned from past experiences under the chosen price. The power at which the policy’s regret grows in the horizon length T would be below (3 +√29)/10 ≃ 0.839 even when demand patterns are fairly ambiguous. When demand realizations are further confined to a finite support, the bound would be reducible to √2/2 ≃ 0.707.

Asymptotically Optimal Lagrangian Policies for One-Warehouse Multi-Store System with Lost Sales

We consider a periodic-review inventory control problem for the Multi-Warehouse Multi-Store system with lost sales. We focus on a time horizon during which the system receives no external replenishment. Specifically, each warehouse has a finite initial inventory at the beginning of the horizon, which is then periodically allocated to the stores in each period in order to minimize the total expected lost-sales costs, holding costs, and shipping costs. This is a hard problem and the structure of its optimal policy is extremely complex. We develop simple heuristics based on Lagrangian relaxation that are easy to compute and implement, and have provably near-optimal performances. In particular, we show that the losses of our heuristics are sublinear in both the length of the time horizon and the number of stores. This improves the performance of existing heuristics in the literature whose losses are only sublinear in the number of stores. Numerical study shows that the heuristic performs very well. We also extend our analysis to the setting of positive delivery lead times.

Forecasting Australia's real house price index: A comparison of time series and machine learning methods

AbstractWe employ 47 different algorithms to forecast Australian log real house prices and growth rates, and compare their ability to produce accurate out‐of‐sample predictions. The algorithms, which are specified in both single‐ and multi‐equation frameworks, consist of traditional time series models, machine learning (ML) procedures, and deep learning neural networks. A method is adopted to compute iterated multistep forecasts from nonlinear ML specifications. While the rankings of forecast accuracy depend on the length of the forecast horizon, as well as on the choice of the dependent variable (log price or growth rate), a few generalizations can be made. For one‐ and two‐quarter‐ahead forecasts we find a large number of algorithms that outperform the random walk with drift benchmark. We also report several such outperformances at longer horizons of four and eight quarters, although these are not statistically significant at any conventional level. Six of the eight top forecasts (4 horizons × 2 dependent variables) are generated by the same algorithm, namely a linear support vector regressor (SVR). The other two highest ranked forecasts are produced as simple mean forecast combinations. Linear autoregressive moving average and vector autoregression models produce accurate olne‐quarter‐ahead predictions, while forecasts generated by deep learning nets rank well across medium and long forecast horizons.

A coupling model of XFEM/peridynamics for 2D dynamic crack propagation and branching problems

Peridynamics (PD) and extended finite element method (XFEM) are coupled in the explicit integration scheme to study the 2D dynamic fracture problems. The advantages of PD for the crack propagation and XFEM for the strong displacement discontinuity description of the crack are combined together. The proposed method can improve the computation efficiency in comparison of the pure PD, and can avoid using macroscopic criteria to judge the crack propagation orientation of XFEM. The coupling scheme for the stiffness and mass matrices of PD and XFEM is given in the coupling region. The horizon length δ is examined to study their influences on the convergence of the proposed method. The accuracy of the dynamic fracture simulation using the proposed method is also validated in comparison of experimental and other numerical methods.

Analytical Aspects of Distributed MPC Based on an Occupancy Grid for Mobile Robots

In this paper, we evaluate theoretical aspects of a distributed system of noncooperative robots controlled by a distributed model predictive control scheme, which operates in a shared space. Here, for collision avoidance, the future predicted state trajectories are projected on a grid and exchanged via discrete cell indexes to reduce the communication burden. The predicted trajectories are obtained locally by each robot and carried out in the continuous space. Therefore, the quantisation does not impose the quality of the solution. We derive sufficient conditions to show convergence and practical stability for the distributed control system by using an idea of a temporary roundabout derived from crossing patterns of street traffic rules, which is established in a fixed and flexible circle size. Furthermore, a condition for the sufficient prediction horizon length to recognise necessary detours is presented, which is adapted for the occupancy grid. The theoretical results match with the trajectory patterns from former numerical simulations, showing that this pattern is naturally chosen as an overall solution.

Sparse and hybrid modelling of relative humidity: the Krško basin case study

This study describes an application of hybrid modelling for an atmospheric variable in the Krsko basin. The hybrid model is a combination of a physics-based and data-driven model and has some properties of both modelling approaches. In the authors’ case, it is used for the modelling of an atmospheric variable, namely relative humidity in a particular location for the purpose of using the predictions of the model as an input to the air-pollution-dispersion model for radiation exposure. The presented hybrid model is a combination of a physics-based atmospherical model and a Gaussian-process (GP) regression model. The GP model is a probabilistic kernel method that also enables evaluation of prediction confidence. The problem of poor scalability of GP modelling was solved using sparse GP modelling; in particular, the fully independent training conditional method was used. Two different approaches to dataset selection for empirical model training were used and multiple-step-ahead predictions for different horizons were assessed. It is shown in this study that the accuracy of the predicted relative humidity in the Krsko basin improved when using hybrid models over using the physics-based model alone and that predictions for a considerable length of horizon can be used.

Real-world cost-effectiveness of rivaroxaban compared with vitamin K antagonists in the context of stroke prevention in atrial fibrillation in France.

ObjectiveThe objective was to assess the real-world cost-effectiveness of rivaroxaban, versus vitamin K antagonists (VKAs), for stroke prevention in patients with atrial fibrillation (AF) from a French national health insurance perspective.MethodsA Markov model was developed with a lifetime horizon and cycle length of 3 months. All inputs were drawn from real-world evidence (RWE) studies: data on baseline patient characteristics at model entry were obtained from a French RWE study, clinical event rates as well as persistence rates for the VKA treatment arm were estimated from a variety of RWE studies, and a meta-analysis provided comparative effectiveness for rivaroxaban compared to VKA. Model outcomes included costs (drug costs, clinical event costs, and VKA monitoring costs), quality-adjusted life-years (QALY) and life-years (LY) gained, incremental cost per QALY, and incremental cost per LY. Sensitivity analyses were performed to test the robustness of the model and to better understand the results drivers.ResultsIn the base-case analysis, the incremental total cost was €714 and the total incremental QALYs and LYs were 0.12 and 0.16, respectively. The resulting incremental cost/QALY and incremental cost/LY were €6,006 and €4,586, respectively. The results were more sensitive to the inclusion of treatment-specific utility decrements and clinical event rates.ConclusionsAlthough there is no official willingness-to-pay threshold in France, these results suggest that rivaroxaban is likely to be cost-effective compared to VKA in French patients with AF from a national insurance perspective.

Sales Effort Management under All-or-Nothing Constraint

We consider a sales effort management problem under an all-or-nothing constraint. The seller will receive no bonus/revenue if the sales volume fails to reach a predetermined sales target at the end of the sales horizon. Throughout the sales horizon, the sales process can be moderated by the seller through her costly effort. We show that the optimal sales rate is non-monotone with respect to the remaining time or the outstanding sales volume required to reach the target. Generally, it has a watershed structure that for any needed sales volume, there exists a cutoff point on the remaining time above which the optimal sales rate decreases in the remaining time and below which it increases in the remaining time. We then study easy-to-compute heuristics that can be implemented efficiently. We start with a static heuristic derived from the deterministic analog of the stochastic problem. With an all-or-nothing constraint, we show that the performance of the static heuristic hinges on how the profit-maximizing rate fares against the target rate, which is defined as the sales target divided by the length of the sales horizon. When the profit-maximizing rate is higher than the target rate, the static heuristic adopting the optimal deterministic rate is asymptotically optimal with negligible loss. On the other hand, when the profit-maximizing rate is lower than the target rate, the performance loss of any asymptotically optimal static heuristic is of an order greater than the square root of the scale parameter. To address the poor performance of the static heuristic for the latter case, we propose a modified resolving heuristic and show that it is asymptotically optimal, and achieves a logarithmic performance loss.

Portfolio selection and unsystematic risk optimisation using swarm intelligence

The financial crisis of 2008, got investment managers to study alternate methods of portfolio construction by focusing on optimisation of risk distribution. In this paper, we introduce a new method of risk diversification of an equity portfolio, based on the individual securities risk contribution. Markowitz’s portfolio optimisation methods were widely believed to diversify the risks until 2010, when it caught the financial market practitioners by surprise. But the scale of misjudgement raised doubts and opened the drawbacks of existing methods in portfolio construction and risk management. In this study, we propose a new technique for portfolio construction and unsystematic risk optimisation, using a well known meta heuristic from evolutionary computation, known as Grey Wolf Optimizer (GWO). We build a portfolio construction model using machine learning methods, to minimise risk contributors from our universe of securities, and then target to increase the asset diversification by maximising the maximum-diversification ratio. We apply boosting techniques to construct portfolio using ensemble of two well known techniques, known as maximum diversification ratio and risk budgeting of portfolio. We have benchmarked our findings by examining the returns of major global indices, and then illustrate our results by comparing with an alternate well known portfolio construction technique, the risk parity portfolio. For comparison, we have considered the length of investment horizon, returns and risks, as factors to present our findings by using the proposed portfolio optimisation and diversification method. Our technique offers practical insights and scope for further investigation into alternate methods of portfolio diversification and risk management using machine learning algorithms.

The contested significance of financial expertise in predicting short- and long-term risk and return on the stock market

This study investigates whether the influence of financial expertise on stock investors’ ability to predict risk and return is contingent on the length of the forecast horizon. In a quasi-experimental design, stock market professionals (N1=63, N2=36), private shareholders (N1=155, N2=172) and students (N1=124, N2=90) twice provided their short- (3-month) and long-term (2-year) risk and return predictions on stock indices. The results show that in general, experts did not outperform students or private shareholders in their return predictions. However, the level of financial expertise positively influenced the accuracy of risk predictions. An interaction effect between financial expertise and the length of the forecast horizon suggests that more knowledgeable and experienced investors performed better in the long term compared to the short term than inexperienced investors did

Cooperation in Indefinite Games: Evidence from Red Queen Games

We test whether the length of an indefinite horizon, or that the horizon is indefinite as opposed to finite, affects cooperation in repeated 3x3 Red Queen Games. With our participant population, and our experimental program and procedures, the length of the indefinite horizon significantly affects cooperation in the Prisoner's Dilemma. However, it does not significantly affect cooperation in the Red Queen Game. Nor does the indefiniteness of the horizon significantly affect cooperation in this latter game. Our results highlight the importance of learning in indefinite games with even slightly larger strategy spaces than the Prisoner's Dilemma.

Efficient Solution of Equality Constrained Quadratic Programming Arising in Model Predictive Control

Fast Model Predictive Control (Fast MPC) is a set of techniques which aim at reducing the complexity of solving receding horizon control optimization problems. One method consists in exploiting the structure induced by the system dynamics. This drastically reduces the complexity of the problem from cubic to linear dependence on the horizon length. Such approach is possible for the multi-stage MPC formulation which is characterized by having an almost separable structure, i.e. coupling occurs only between consecutive stages. This paper proposes a novel technique that efficiently solves general linear-quadratic optimal control problems and hence does not require a multi-stage formulation. The method also provides other advantages including: higher solution accuracy, warm starting, and approximate solutions by early termination. This is achieved by embedding the Riccati recursion in a Projected Conjugate Gradient (PCG) method.

Solution for the Continuous-Time Infinite-Horizon Linear Quadratic Regulator Subject to Scalar State Constraints

This letter provides a solution for the continuous-time linear quadratic regulator (LQR) subject to a scalar state constraint. Using a dichotomy transformation, novel properties for the finite-horizon LQR are derived; the unknown boundary conditions are explicitly expressed as a function of the horizon length, the initial state, and the final state or cost of the final state. Practical relevance of these novel properties are demonstrated with an algorithm to compute the continuous-time LQR subject to a scalar state constraint. The proposed algorithm uses the analytical conditions for optimality, without a priori discretization, to find only those sampling time instances that mark the start and end of a constrained interval. Each subinterval consists of a finite-horizon LQR, hence, a solution can be efficiently computed and the computational complexity does not grow with the horizon length. In fact, an infinite horizon can be handled. The algorithm is demonstrated with a simulation example.